An optimal control problem in polyconvex hyperelasticity

We consider a shape implant design problem that arises in the context of facial surgery.
We introduce a reformulation as an optimal control problem, where the control acts
as a boundary force. The state is modelled as a minimizer of a polyconvex
hyperelastic energy functional. We show existence of optimal solutions and
derive - on a formal level - first order optimality conditions. Finally, preliminary numerical results
are presented.